Schoool of Economics and Commercial Law Industrial and Financial Program
Supervisor: Wilhborg Clas
Authors: Hu Pengfei Hua Yimin
Real Option Valuation Real Option Valuation
in High
in High--Tech Firm Tech Firm
Abstract
In traditional financial theory, the discounted cash flow model (or NPV) operates as the basic framework for most analyses. In doing valuation analysis, the conventional view is that the net present value (NPV) of a project is the measure of the value that is the present value of expected cash flows added to the initial cost. Thus, investing in a positive (negative) net present value project will increase (decrease) value.
Recently, this framework has come under some fire for failing to consider the options which are the managerial flexibilities, which are the collection of opportunities.
A real-option model (Option-based strategic NPV model) is estimated and solved to yield the value of the project as well as the option value that is associated with managerial flexibilities. Most previous empirical researchers have considered the initial-investment decision (based on NPV model) but have neglected the possibility of flexible operation thereafter. Now the NPV must be compared with the strategic option value, by which investment is optimal while the NPV is negative. This leads investors to losing the chances to expand themselves.
In the paper we modify the NPV by taking into account real options—
theme of this paper, or strategic interactions. R&D, Equity and Joint Ventures will be viewed as real options in practice of case studies of this paper.
Keywords: Discount rate, Net present value (NPV), Option(s) and valuation
Acknowledgments
Those persons who have inspired us, helped us, or corrected us in the course of time are too numerous to be named. Thus we will thank here only Clas Wilhborg, our supervisor, who spent his invaluable time in inspiring us.
We are grateful to Gert Sandahl and Matin Hómen who made helpful contributions to both our studies and this paper.
For great help in running this program as significant guide for our studies, we owe a special debt to Ann McKinnon.
Finally, our warmest thanks to those who develop the original drafts of this paper, who correct the grammar, who stand by us enduring late nights but encouraging us.
We are talking in a vacuum.
We are talking in a void.
We are talking in a state of quantal bliss.
Because everything’s uncertain We can’t even tell the time.
It’s a proble* that defies analysis
---B.K. Ridley
* Proble—a problem without end
Table of Contents
Abstract ...i
Acknowledgments ... ii
Part I: Static Tools for Valuation ...1
1 Introduction ...1
1.1 Background...2
1.2 Problems Identification ...3
1.3 Objectives ...4
1.4 Limitations...4
1.5 Methodology ...5
2 Traditional Valuation Method ...5
2.1 Major Approaches to Determine the Discount Rate...6
2.1.1 Capital Asset Pricing Model (CAPM)...6
2.1.2 Weighted Average Cost of Capital (WACC) Approach ...8
2.2 DCF-NPV Approach to Project Valuation...9
2.3 DCF-NPV Approach to Firm Valuation ...10
2.4 Failure of DCF-NPV Method ...12
Part II: Expanding of Financial Option to Real Option...16
3 Financial-Options and Option-Pricing Theory ... 16
3.1 Basic Concepts of Options ...17
3.2 Pricing for Options ...20
3.3 Option Pricing Models...24
3.3.1 The Binomial Model ...24
3.3.2 Extending the Binomial Model to Continuous Time—Black-Scholes Option Pricing Model ...28
3.3.3 Comparison between Binomial Model and Black and Scholes Model...32
4 Real Options ... 35
4.1 Survey of Real Option ...35
4.1.1 The Option to Defer a Project ...36
4.1.2 The Option to Expand a Project ... 37
4.1.3 The Option to Abandon a Project ...38
4.1.4 Managerial Flexibility, Asymmetry, and Strategic (Expanded) NPV ...38
4.2 Inputs for Real Options ...40
4.3 Valuing a Firm as a Real Option...41
4.3.1 Stocks and Bonds ...41
4.3.2 M&M vs. Option Approach...44
4.3.3 About the Assumptions of Option Approach on Equity Valuation ...46
4.3.4 Other Corporate Financial Claims ...47
4.4 Compound Options Correspond to Interrelated Projects ...51
4.5 Advantage and Disadvantage of Real Option Approach ...52
4.5.1 Advantage of Real Option Approach ...52
4.5.2 Disadvantages of Real Option Approach ...53
Part III: Implementation of Real Options...55
5 Case Studies... 55
5.1 Projects as Real Options ...56
5.1.1 Valuation for R&D...56
5.1.2 Valuation for Equity (Capital Structure) ...60
5.2 Joint Ventures as Real Options...65
5.2.1 Microsoft and Ericsson Alliance on Wireless Access... 65
6 Conclusions ... 69
References ...71
1. Books... 71
2. Articles ... 72
3. Internet... 73
Appendices ...74
1. The Basic Valuation Idea: Risk-Neutral Valuation... 74
2. Calculation for Volatility ... 74
3. Cumulative Probability for the Standard Normal Distribution... 76
Content of Figures
Figure 3.1 Binomial Trees from Specific Case to General Case ...25
Figure 3.2 Stock and Option Prices in General Two-step Tree ...26
Figure 3.3 Payoffs from Positions in European Options ...34
Figure 4.1 The Option to Defer a Project ...36
Figure 4.2 The Option to Expand a Project ...37
Figure 4.3 The Option to Abandon a Project ...38
Figure 4.4 Managerial Flexibility or Options Introduce an Asymmetry into the Probability Distribution of NPV... 39
Figure 4.5 Payoffs on Equity as Option on a Firm...43
Figure 4.6 Value of Warrant ...48
Figure 4.7 Valuations for Convertible Bonds ... 50
Figure 5.1 Payoffs from R&D...57
Figure 5.2 Calculations for Strategic NPV on R&D ...59
Figure 5.3 Calculations for Euro-Tunnel Case ... 62
Figure 5.4 Calculations for Ericsson Equity 1998 ...64
Content of Tables Table 2.1 Comparisons between CAPM and M&M ...12
Table 3.1 Summary of Factors Affecting Call and Put Prices ...24
Table 3.2 The Binomial Option Pricing Model ...28
Table 3.3 Summary for Modified Black-Scholes Model...31
Table 3.4 Payoffs from Positions in European Options ...34
Table 4.1 Comparison of Inputs between a Call Option and a Real Option...41
Table 5.1 Euro-Tunnel Debt Value and Duration ...61
Table 5.2 Ericsson Debt Value and Maturity Period...64
Framework for the paper:
Static analyzing
Project
Dynamic analyzing Traditional NPV
Option-based strategic NPV model
Real options valuation
CAPM WACC
Financial options
Defer Abandon
Expand
Case Studies MNC
Exchange rate
Traditional Tech.
M&M Ad hoc occurrence
Decision-making process Discounted rate
Cash flows
Part I: Static Tools for Valuation 1 Introduction
If we must rethink in this paper the links between customary methods of valuation and the ones emerging currently, it is because we humans evolve from one stage to another.
People have found that there are some pitfalls in most traditional methods of valuation, NPV for example. Our objective is to introduce a new way to valuation, or in a certain sense repair traditional NPV to be strategic option- based NPV method.
It is thus necessary first to go back to the traditional valuation methods. Only then, against this background, may we be able to appreciate the evolution of knowledge in the course of time and see the turning point of valuation methods. This is precisely what we are going to do.
We shall first talk about static valuation tools, NPV which involves with discounted rate, cash flows, and so on. To which a couple of magnificent theories, CAPM and WACC will be covered.
In view of the static and passive feature of NPV, we shall bring in the more flexible financial option theory. And by combining financial option with NPV, an option-based NPV—Real Option theory might well appear to us. This above is just the second part of this paper: the expanding of financial option to real option.
For deeply understanding real options, the implementation of real option will be scrutinized in third part of the paper.
1.1 Background
Traditional financial theory states that the value of a firm is independent of its capital structure in the absence of taxes and bankruptcy. It has been demonstrated true in 4 prospects:
Ø The returns on the underlying assets are certain and investors can borrow at the risk free rate.
Ø The firm is part of an efficient market.
Ø There are two or more firms with different leverage in the same “ risk class”, and these firms will no go bankrupt with probability
Ø Investors can arrange “no course” loans with the stock as collateral (Ingeroll, 1987)
The main approach to value firm by traditional financial theory is discounted cash flow (DCF) model. DCF models are used for project evaluation by most companies, presumably because they are straightforward to apply and because they are intuitively appealing. One needs to forecast the future cash flows, choose the appropriate discount rate, and find the present value of the forecasted cash flows. The net present value (NPV) is defined as the difference between the present value of the future cash flows and the initial cost. If NPV is positive, then accepting the project adds value to firm. Given accurate estimates of future cash flows, the success of the discounted cash flow then will depend on how well you choose the discount rate. If you pick a rate that is too high, you will reject projects that have negative NPV; if you pick a rate that is too low, you will accept projects that have positive NPV.
However the DCF model values the firm with a certain deterministic discount rate that of course is unrealistic in an uncertain world. It ignores the value of management, growth, deferring, liquidation and abandonment value in real assets and other factors that may impact the value of the firm.
Recent financial literature points out that a firm’s equity, debt and even itself can be treated as a real option for investors. For example, the implication of viewing equity as a call option is that equity will have value, even if the value of the firm falls well below the face value of the outstanding debt. While the firm will be viewed as troubled by investors, accountants, and analysts, its equity is not worthless. In fact just as deep out-of-the-money traded options command value because of the possibility that the value of the underlying asset may increase above the strike price in the remaining lifetime of the option, equity commands value because of the time premium on the option and the possibility that the value of the assets may increase above the face value of the bonds before they come due. Thus the real option approach to firm valuation can explain why highly levered, risky firms still have a high value in equity markets. Not only explaining by general option theory, but also the real option approach may derive value from the firm equity for relevant investors.
1.2 Problems Identification
The problem of this paper is mainly of examining the real option theory in both theory and case studies, to which strategic decision-making based on real options could presumably maximize the wealth of the shareholders of a corporation. In doing so, we shall divert our analyses from traditional static viewpoints to dynamic ones that integrate managerial flexibility (uncertainty) for instance, which were neglected in traditional valuation tools.
In theory part, we review proved academic research on real option. While in case studies, we give the quantitative analysis and theoretical explanations to the business events.
1.3 Objectives
One vital consequence of uncertainty is that the laws of social science have to be statistical in character. Real Option theory has to talk about expectation values rather than determined quantities, probabilities rather than certainties.
Prediction is more uncertain, however, real option theory does quantify our model precisely, which can be remarkable.
The goal of this thesis is mainly about applying real options to real case, for example Ericsson Radio System. In other words, we want to make the above financial theories more applicable in life by transforming numerical outcomes into strategic decision making for a corporation.
1.4 Limitations
The thesis work is rather a theoretical one. Most parts of the thesis are the review of the empirical studies.
The results of case study are actually calculated upon some assumed inputs that certainly may cause some biases from real outcomes. Assumptions are made consistent with NPV method for the estimation of future value.
The interactive strategy of game theory is excluded.
1.5 Methodology
By comparing the different tools, we illustrate the advantages and disadvantages of different approaches, traditional NPV and Option-base strategic NPV, to valuation. The above approaches will be used to implement the objectives of the paper. Very beneficial for both authors and readers is to speak of two sorts of ways for contemplating the Real Options in strategic NPV valuation: Binomial trees and Black-Scholes methods. Traditional NPV will be improved for measuring the more flexible underlying assets, for instance a corporation. Strategic NPV tries to reasonably estimate a project (firm) for decision-makers.
2 Traditional Valuation Method
It is nearly impossible to discuss valuation without uncertainties. Valuation approach under certainty circumstances only exists in treasure issues buying.
Investors may enjoy a risk free rate on their holding these kinds of issues. A simple discounted cash flow (DCF) model can be well used for certainty valuation.
However in the real world, uncertainties do affect the value. Bierman and Smidt (1993) pointed out that the traditional technique for dealing with uncertainty could be classified into two groups. One group of techniques attempts to consider explicitly all alternative sequences of cash flow. The state preference approach fits in this group. These techniques are attractive theoretically but are difficult to implement. The commonly used practical techniques are often methods of approximating the result that would be obtained if a theoretically correct approach were used.
A second group of techniques requires the decision maker to provide a concise summary description of the asset that can be used to make an estimate of its value. For example, the decision maker may estimate the expected cash flows of each period and discount these by an appropriate risk adjusted discount rate to estimate the value of the asset. In estimating values for bond, the promised (most likely) cash flows are used in place of the expected cash flows. In the capital asset pricing model (CAPM), it is assumed that the decision maker knows the asset’s beta coefficient, which describes the relationship between the values of the asset (or of some closely related asset) following a particular probability distribution and must specify parameters of the distribution, such as its variance. With certainty equivalent approach, the uncertain cash flows of each period are collapsed into a single measure that reflects both probabilities and risk preferences. All of these techniques aim to produce an estimated market valuation for the investment proposal. This group approach is actually the extension of discounted cash flow- net present value (DCF-NPV) analysis.
This group approach is widely used in today’s capital valuation, probably because they are straightforward to apply and because they are intuitively appealing. You forecast the cash flows, choose the appropriate discount rate, and find the present value of the forecasted cash flows. If the NPV is positive, then accepting the project adds value to the firm.
We would like to introduce two approaches determining the discount rate before we have a review on DCF-NPV method in valuation.
2.1 Major Approaches to Determine the Discount Rate
2.1.1 Capital Asset Pricing Model (CAPM)
CAPM is an equilibrium model of asset pricing that states that the expected return on a security is a positive linear function of the security’s sensitivity to changes in the market portfolio’s return. (Sharpe, 1997) The key variable in
the CAPM is called “beta”, a statistical measure of risk which has become as familiar as—and, indeed, interchangeable with—the CAPM itself. Financial managers have long realized that some projects were riskier than others, and that these projects require a higher rate of return. A risky investment is, of course, one whose return is uncertain in advance; and in such a case, it is only the expected or average rate of return that can be projected. To justify undertaking the risky project, a higher payout in the event of success is required.
The simple CAPM Model captures this perspective. According to the simple CAPM, an investment’s required rate of return increases in direct proportion to its beta. The CAPM also implies that investors, in pricing common stocks, are concerned exclusively with systematic risk. A security’s systematic risk, as measure by beta, is the sensitivity (or co-variance) of its return to movements in the economy as a whole. Asset with high betas exaggerate general market developments, performing exceptionally well when the market goes up and exceptionally poorly when the market goes down (Rosenburg and Rudd, 1998).
The simple CAPM model states:
[
m f]
βf ER R
R R
E( )= + ( )−
where: E (R) = The required rate of return (or rate of return)
Rf = The Risk-free rate (the rate of return on a “risk-free investment”, like U.S. government treasury bonds)
β = Beta (see above)
E (Rm) = the expected return on the overall stock market
In other words, the required rate of return is equal to the sum of two terms: the risk-free return and an increment that compensates the investor for accepting the asset’s risk. The compensation for risk is expressed as the asset’s beta multiplied by the expected excess return of the market, [E (Rm) – Rf]. This expected excess return is sometimes referred to as the “risk premium”(Rosenburg and Rudd, 1998).
2.1.2 Weighted Average Cost of Capital (WACC) Approach
The cost of capital to a firm may be defined as a weighted average of the cost of each type of capital. The weight of each type of capital is the ratio of the market value of the securities representing that source of capital to the market value of all securities issued by the company. The term security includes common and preferred stocks and all interest-bearing liabilities, including notes payable. It is sometimes stated that the weighted average cost capital of a firm may be used to evaluate investments whose cash flows are perfectly correlated with the cash flows from the firm’s present assets. With perfect correlation between the two sets of cash flows, the risk is the same (Bierman and Smidt, 1993). The usual definition of the weighted average cost of capital is to weight the after-tax cost of debt by the percentage of debt in the firm’s capital structure and add the result to the cost of equity multiplied by the percentage of equity. The equation is
S B K S C B K B
WACC b c s
+ +
− +
= (1 τ )
The cost of capital combines in one discount rate an allowance for the time value of money and an allowance for risk. To apply the same cost of capital to cash flows that occur at different points in time, the magnitude of these allowances (i.e., the percent per unit of time) must remain constant over time.
A specific asset might have a smaller or larger amount of risk, thus should have a smaller or larger discount rate. The WACC is the correct discount rate only for one level of risk. For a given capital structure, the weighted average cost of capital to a firm reflects the characteristics of the firm’s assets, and particularly their average risk, but also the timing of the expected cash proceeds.
Both CAPM and WACC approaches can be used to determine discount rate for valuing the future cash flow to the project and firm.
2.2 DCF-NPV Approach to Project Valuation
Most capital-budgeting investments, of course, involve discounting of cash flow over multiple future periods, so that using the single period CAPM for discounting one period at a time would necessitate certain additional assumptions concerning the evolution of its variables over time. Fama (1977 shows that the present value of a future net cash flow is its current expected value discounted at risk-adjusted discount rates Ks given by the CAPM. The discount rates must be known and non-stochastic (i.e., they must evolve in a deterministic fashion through time), and in general they will differ from period to period (and across cash flows for a given period). The NPV of an investment project is then simply the sum of the present values of all its future net cash flows.
[ ]
∑
= += N
t
t s
jt
j K
NPV NCF
0 1
2.3 DCF-NPV Approach to Firm Valuation
Modigliani and Miller (1958, 1963) wrote a breakthrough working paper on cost of capital, corporate valuation, and capital structure based listed assumptions1:
Ø Capital markets are frictionless.
Ø Individuals can borrow and lend at the risk-free rate.
Ø There are no costs to bankruptcy.
Ø Firms issue only two types of claims: risk free debt and (risky) equity.
Ø All firms are in the same risk class.
Ø Corporate taxes are the only form of government levy (i.e., there are no wealth taxes on corporations and no personal taxes)
Ø All cash flow streams are perpetuities (i.e., no growth)
Ø Corporate insiders and outsiders have the same information (i.e., no signaling opportunities).
Ø Managers always maximize shareholder’s wealth (i.e., no agency costs).
Although many of these assumptions are unrealistic, they do not really change the major conclusions of the model of firm behavior (Copland and Weston, 1992).
Suppose the assets of a firm return the same distribution of net operating cash flow each time period for an infinite number of time periods. This is a no growth situation because the average cash flow does not change over time.
The value of the firm can be written as below:
1 See Copeland/Weston “Financial Theory and Corporate Policy” third edition P439, Addison-Wesley Publishing Company.
ρ ) (FCF VU =E
where: VU = the present value of an unlevered firm (i.e., all equity) E (FCF) = the perpetual free cash flow after taxes ρ = the discount rate for an all-equity firm of equivalent risk.
This is the value of an un-levered firm because it represents the discounted value of a perpetual, non-growing stream of cash flows after taxes that would accrue to shareholders if the firm had no debt.
To value a levered firm, the equation below can be derived from M&M approach.
WACC VL= NOI
where:VL=The present value of a levered firm NOI= Net Operating Income
The discount rate for NPV which determines the value of the (un-levered and levered) firm can be found in Table 2.1:
Table 2.1 Comparisons between CAPM and M&M (Source: Copeland and Weston 1992) Type of
Capital
CAPM Definition M&M Definition
Debt Kb=Rf+
[
E(Rm)−Rf]
βb Kb=Rf,βb=0Un-levered
Equity ρ=Rf+
[
E(Rm)−Rf]
βu ρ=ρLevered
Equity Ks Rf ERm Rfβs
−
+
= ( )
S K B Ks=ρ+(ρ− b)(1−τc) WACC for
the firm WACC=
S B K S C B
Kb c B + s +
− ) + 1
( τ WACC= (1 )
S B
B
c +
−τ ρ
2.4 Failure of DCF-NPV Method
The application of the DCF-NPV to the valuation of real risky assets is made possible by two almost tacit assumptions or conventions. The first is that uncertain future cash flow can be replaced by their expected values and that these expected cash flows can be treated as given at the outset. The second is that the discount rate is known and constant, and that it depends solely upon the risk of the project. Let us consider the limitations of an approach based on these assumptions and see why the underlying DCF-NPV analogy may be a poor one for some investment projects.
First, by assuming that the cash flows to be discounted are given at the outset, pre-supposes a static approach to investment decision-making—one which ignores the possibility of future management decisions that will be made in response to the market considerations encountered. Over the life of a project, decisions can be made to change the output rate (ad hoc discounted rate should be used), to expand or close the facility, or even to abandon it. The
flexibility afforded by these decision possibilities may contribute significantly to the value of the project.
To introduce an analogy which we shall develop further below, the DCF-NPV approach may be likened to valuing a stock option contract while ignoring the right of the holder not to exercise when it is unprofitable. To some extent this drawback of the DCF-NPV approach may be overcome by employing a scenario or simulation approach in which alternative scenarios-involving for example different price outcomes and management responses-are generated and the resulting cash flows estimated. These cash flows are then averaged across scenarios and discounted to arrive at the present value.
Unfortunately this scenario or simulation approach gives rise to two further problems. First, it requires that the appropriate policy for each scenario be determined in advance. Of course sometimes this will be possible. For example, if the output rate can be adjusted without any costs, the simple rule of setting marginal cost equal to price may sometimes be optimal. But more generally this will not be possible. If it is costly to close or abandon a project, then the decision to close is itself an investment decision with uncertain future cash flows depending on commodity prices. The optimal closure policy must therefore be determined simultaneously with the original capital budgeting decision.
Even more fundamentally, the degree of managerial discretion in making future operating decisions will tend to affect the risk of the project under consideration. A project which can be abandoned under adverse circumstances will be less risky than one that cannot; it will be even less risky if part of the initial capital investment can be recovered in the event of abandonment. The classical approach offers no way of allowing for this risk effect except through some ad hoc adjustments of the discount rate.
In fact the tacit assumption concerning the discount rate is the second Achilles’ heel of the classical approach. Given any set of expected cash flows, there almost always exists some discount rate, which will yield the correct present value. But the determination of this discount rate presents a quite difficult task, and current procedures cannot be regarded as any more than highly imperfect rules of thumb. Thus these procedures all assume that the discount rate is constant, which is equivalent to assuming that the risk of the project is constant over its life. And this is, of course, highly unlikely. Not only will the risk depend in general upon the remaining life of the project, it will almost certainly depend upon the current profitability of the project through an operating leverage effect. Hence, not only will the discount rate vary with time, it will also be uncertain.
Even if the appropriate discount rate were deterministic and constant, the problem of estimation would still be formidable. In principle the discount rate should depend upon the risk of the project, but how is this risk to be assessed?
The generally approved procedure is to use the CAPM and to base the discount rate on the beta of the project as estimated from other firms with similar projects. In practice these other firms consist in effect of portfolios of projects, sometimes in unrelated industries, and this makes the assignment of betas to individual projects a hazardous undertaking. Transferring these betas to the project under consideration creates further problems, for a new project is likely to have a cost structure that differs in a systematic fashion from existing, mature projects. The problem is compounded by the consideration, mentioned above, that the latitude of future operating decisions inherent in a project will affect its risk, and is unlikely to be duplicated in existing projects.
Of course these problems are often ignored in practice and a single corporate discount rate based on the weighted average cost of capital is employed for all projects, regardless of risk. As is well known, however, the price of this simplification is a capital budgeting decision system which contains
systematic biases as between projects with different risks and different lives.
And, such a decision system will lead to the systematic under-valuation of projects with significant operating options.
A final practical difficulty with the classical approach is the necessity to forecast expected output prices for many years into the future. A wide range of possibilities for the path of expected future spot prices will appear plausible, and the calculated present value of the project will depend upon some arbitrary selection among them.
The above appears to constitute a fairly strong indictment of the classical discounted cash flow approach to capital budgeting. The limitations of the classical approach arise because it is based fundamentally on an analogy between a portfolio of risk-less bonds and a real investment project. In many cases this analogy may be useful; for example, in situations in which the scope for future managerial discretion is limited, and the fiction of other similar risk projects can be maintained.
For improving the static feature of traditional DCF-NPV approach, option theory could be taken into account, from which investor can really find an
“option” to implement his idea.
Part II: Expanding of Financial Option to Real Option
The discounted cash flow method is a powerful and valuable tool for valuing such projects as routine projects, or projects that involve replacement of existing machinery with more efficient equipment.
However, many other projects require considerable intervention (flexibility) from management of a firm (project). The application of this type of intervention in management increases more opportunities for project (firm).
Therefore, it is quite important to take these opportunities (options) into account when you value a project.
As indicated by the title of this part, we shall expand financial option theory to real option theory in this part.
3 Financial-Options and Option-Pricing Theory
A financial derivative security (derivative) is an instrument whose value depends on the price of its underlying variables, including stocks, stock indices, foreign currencies, debt instruments, commodities and future contracts and so on. The derivatives, (forward contracts, swaps, and options for instance), are also known as contingent claims for they can actually solve the ad hoc problems.
In this paper we mainly concentrate on options. A stock option for example, is a contract which conveys to its holder the right, but not the obligation, to buy or sell shares of the underlying asset at a specified price on or before a given date. This right is granted by the seller of the option.
3.1 Basic Concepts of Options2
In financial markets people may very often hear one word: options. Two types of options are in general used for either hedging or speculating: puts and calls, which are contracts that give the owner the right (no obligation) to do something. The options’ contracts are different from futures. Just for option, it offers the holder of option the right to do something, leading to the prices being imposed on them (Option premier).
A call option is the right for holder of the option to buy the Underlying asset by a certain date at the pre-negotiated price. A put option is the right for holder of the option to sell the underlying asset by a certain date at the pre- negotiated price. For example, an American-style XYZ Corp. May 60 call entitles the buyer to purchase 100 shares of XYZ Corp. common stock at $60 per share at any time prior to the option’s expiration date in May. Likewise, an American-style XYZ Corp. May 60 put entitles the buyer to sell 100 shares of XYZ Corp. common stock at $60 per share at any time prior to the option’s expiration date in May.
Underlying Asset: The specific asset on which an option contract is based is commonly referred to as the underlying assets. The underlying of an option can be any asset at all, as long as it has a value upon which both sides of the contract can agree (Chriss, 1997). For example, the underlying can be a commodity (gold or silver), or a foreign currency (U.S. dollar-yen exchange rate), or stock indexes (Standard & Poor index) and so on. Options are categorized as derivative securities because their value is derived in part from the value and characteristics of the underlying asset (security for instance). A stock option contract’s unit of trade is the number of shares of underlying stock which are represented by that option. Generally speaking, stock options
2 Source: Chicago Board Option Exchange, Feb 1999.
have a unit of trade of 100 shares. This means that one option contract represents the right to buy or sell 100 shares of the stock (underlying asset).
Strike price: The strike price, or exercise price, of an option is the specified underlying asset price at which the underlying asset can be bought or sold by the holder, or buyer, of the option contract if he/she exercises his/her right against a writer, or seller, of the option. To exercise your option is to exercise your right to buy (in the case of a call) or sell (in the case of a put) the underlying asset at the specified strike price of the option.
The strike price for an option is initially set at a price which is reasonably close to the current underlying asset price. Additional or subsequent strike prices are set at the following intervals: 2½ -points when the strike price to be set is $25 or less; 5-points when the strike price to be set is over $25 through
$200; and 10 points when the strike price to be set is over $200. (New strike prices are introduced when the price of the underlying asset rises to the highest, or falls to the lowest, strike price currently available). The strike price, a fixed specification of an option contract, should not be confused with the premium, the price at which contract trades, which fluctuates daily.
If the strike price of a call option is less than the current market price of the underlying asset, the call is said to be in-the-money because the holder of this call has the right to buy the stock at a price which is less that the price he would have to pay to buy the stock, for example, in the stock market.
Likewise, if a put option has a strike price that is greater than the current market price of the underlying asset, it is also said to be in-the-money because the holder of this put has the right to sell the stock at a price which is greater than the price he would receive selling the stock in the stock market. The converse of in-the-money is, not surprisingly, out-of-the-money. If the strike price equals the current market price, the option is said to be at-the-money.
Premium: Option buyers pay a price for the right to buy or sell the underlying asset. This price is called the option premium. The premium is paid to writer, or seller, of the option. In return, the writer of a call option is obligated to deliver the underlying asset to a call option buyer if the call is exercised.
Likewise, the writer of a put option is obligated to take delivery of the underlying asset from a put option buyer if the put is exercised. Whether or not an option is ever exercised, the writer keeps the premium. Premiums are quoted on per underlying unit basis. Thus, a premium of 7/8 represents a premium payment of $87.50 per option contract ($0.875 • 100 units).
American, European and Capped styles: There are 3 styles of options:
American, European and Capped. In the case of an American option, the holder of an option has the right to exercise his option on or before the expiration date of the option. A European option is an option which can only be exercised during a specified period of time prior to its expiration. A capped option gives the holder the right to exercise that option only during a specified period of time prior to its expiration, unless the option reaches the cap value prior to expiration, in which case the option is automatically exercised. The holder or writer of either style of option can close out his position at any time simply by making an offsetting, or closing, transaction. A closing transaction is the transaction in which, at some point prior to expiration, the buyer of an option makes an offsetting sale of an identical option, or the writer of an option makes an offsetting purchase of an identical option. A closing transaction cancels out an investor’s previous position as the holder or writer of the option.
The Option Contract: An option contract is defined by the following elements: type (put or call), style (American, European and Capped), underlying asset, unit of trade (number of shares), strike price, and expiration date. All option contracts that are of the same type and style and cover the same underlying asset are referred to as a class of options. All options of the
same class that also have the same unit of trade at the same strike price and same expiration date are referred to as an option series.
If a person’s interest in a particular series of options is as a net holder (that is, if the number of contracts bought exceeds the number of contracts sold), then this person is said to have a long position in the series. Likewise, if a person’s interest in a particular series of options is as a net writer (if the number of contracts sold exceeds the number of contracts bought), he is said to have a short position in the series. After making contract, the thing people face is to exercise the option.
Exercising the option: if the holder of an option decides to exercise his right to buy (in the case of a call) or to sell (in the case of a put) the underlying asset, the holder must direct his broker to submit an exercise notice to OCC (Options Clearing Corporation). In order to ensure that an option is exercised on a particular day, the holder must notify his broker before the broker’s cut- off time for accepting exercise instructions on that day.
Based on these basic concepts, we can proceed more easily to the next section:
pricing the options.
3.2 Pricing for Options
There are several factors that contribute value to an option contract and thereby influence the premium or price at which it is traded. The most important of these factors are the prices of the underlying assets; time remaining until expiration, the volatility3 of the underlying asset price, cash dividends and interest rates.
3 See Appendix 2
Factors affecting option value4: The value of an option is determined by a number of factors relating to the underlying asset and financial markets.
1) Underlying Asset Price: the value of an option depends heavily upon the price of its underlying asset. As previously explained, if the price of the stock is above a call option’s strike price, the call option is said to be in-the- money. Likewise, if the stock price is below a put option’s strike price, the put option is in-the-money. The difference between an in-the-money option’s strike price and the current market price of a share of its underlying security is referred to as the option’s intrinsic value. Only in-the-money options have intrinsic value.
For instance, if a call option’s strike price is $45 and the underlying shares are trading at $60, then the option has intrinsic value of $15 because the holder of that option could exercise the option and buy the shares at $45. The buyer could then immediately sell these shares on the stock market for $60, yielding a profit of $15 per share, or $1,500 ($15*100) per option contract.
When the underlying share price is equal to the strike price, the option (either call or put) is at-the-money. An option that is not in-the-money or at-the- money is said to be out-of the-money. An at-the-money or out-of-the-money option has no intrinsic value, but this does not mean it can be obtained at no cost. There are other factors that give options value and therefore affect the premium at which they are traded. Together, these factors are termed time value. The primary components of time value are time remaining until expiration, volatility, dividends, and interest rates. Time value is the amount, by which the option premium exceeds the intrinsic value, i.e.,
Option Premium = Intrinsic Value + Time Value
4 (Source: Chicago Board Option Exchange, Feb, 1999)
For in-the-money options, the time value is the excess portion over intrinsic value. For at-the-money and out-of-the-money options, the time value is the total option premium.
2) Time Remaining Until Expiration: Generally, the longer the time remaining until an option’s expiration date, the higher the option premium because there is a greater possibility that the underlying asset price might move so as to make the option in-the-money. Time value drops rapidly in the last several weeks of an option’s life.
3) Volatility: Volatility is the propensity of the underlying security’s market price to fluctuate either up or down. Therefore, volatility of the underlying asset price influences the option premium. The higher the volatility of the stock, the higher the premium because there is, again, a great possibility that the option will move in-the-money.
4) Dividends paid on the Underlying Assets: Regular cash dividends are paid to the stockholder. Therefore, cash dividends affect option premiums through their effect on the underlying asset (stock/share) price. Because the stock price is expected to fall by the amount of the cash dividend, higher cash dividends tend to imply lower call premiums and higher put premiums.
Options customarily reflect the influences of stock dividends (e.g., additional shares of stock) and stock splits because the number of shares represented by each option is adjusted to take these changes into consideration.
5) Interest Rates: Historically, higher interest rates have tended to result in higher call premiums and lower put premiums.
Premiums (prices) for exchange-traded options are published daily in a large number of newspapers. A typical newspaper listing, as below, can give us a better understanding of option premium:
Calls-Last Puts-Last
Option &
NY Close Strike
Price 5) May Jun Jul May Jun Jul
1) XYZ 3) 105 4) 7½ 9 1 4 10 1 4 1
2)112 110 3 4 3 4 6 1 4 1 3 16 1 2
112 115 13 16 2 3 1 2 4 4 5
112 120 3 16 1 3 4 8 8 8 3 4
112 125 1 16 s 13 16 r s r
112 130 s s s s 18 3 4
1) stock identification 4) closing option prices
2) stock closing price 5) option expiration months
3) option strike prices r = not traded s = no option listed (Source: Chicago Board Option Exchange, Feb, 1999)
In this example, the out-of-the-money XYZ July 115calls closed at 3 1 2, or
$350 per contract, while XYZ stock closed at 112 . The in-the-money July 120 puts closed at 8 3 4, or $875 per contract (Understanding stock options, Feb 1999, Chicago Board Option Exchange).
These factors and their predicted effects on call and put prices are also summarized below:
Table 3.1 Summary of Factors Affecting Call and Put Prices Effect on Call Value Put Value underlying asset’s value
strike price volatility (variance) time to expiration interest rates dividends paid
3.3 Option Pricing Models
In this section, we introduce the most widely used model for options pricing—
the binomial model. This model for option pricing is much more generalized than the Black-Scholes model which based on the geometric Brownian motion model. The former is a discrete time model while the latter is a continuous time one.
3.3.1 The Binomial Model
A paper entitled “Option Pricing: A Simplified Approach” appeared in 1979.
Basically, the development of the binomial model was spurred by an attempt by Professors Cox, Ross and Rubinstein to find an easier way to teach their students how the Black and Scholes formula (which we will have an explanation in later chapters) works. Just they introduced the Binomial model to option pricing. And Binomial model is an extremely powerful tool for pricing a wide variety of options. So popular and very widespread a tool it is in today’s world.
The binomial model has proved over time to be the most flexible, intuitive and popular approach to option pricing. It is based on the simplification that over a
Factors
Increase in
single period (of possibly very short duration), the underlying asset price can only move from its current price to two possible levels. The general formulation of a stock price process that follows the binomial is shown in Figure 3.1 below.
Figure 3.1 Binomial Trees from Specific Case to General Case (General Formulation for Binomial Price Path)
In Figure 3.1, S0 is the current stock price; the price moves up to Su with probability p and down to the Sd with probability (1 – p) in any time period.
For details on binomial formulation, it is necessary and convenient to bring in the Replicating portfolio used by Black and Scholes, which is a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued. By following this “replicating portfolio” Black and Scholes came up with their final formulation, as we will see later. We introduce this portfolio here to derive the binomial formulation.
The objective of establishing a replicating portfolio is to combine risk-free Su
95.12 102.5 107.7
92.77 97.53
110.5
105.1
95.12
90.48 105.1
100.0 100.0
97.53 102.5
100.0 S0
Sd
S ud
borrowing/lending and the underlying asset, with which we can create the same cash flows as the option being valued.
The tree in Figure 3.1 is named a recombining tree, of which the most important property is that, an up move followed by a down move is at any time exactly the same as a down move followed by an up movement (Chriss, 1997). We, in this paper, will mainly study these sorts of trees. The only assumption we need here is that there are no arbitrage opportunities for an investor. In terms of the definition of the replicating portfolio, the value of the option is equal to the value of the replicating portfolio. The replicating portfolio for a call with strike price K will involve borrowing $B and acquiring of the underlying asset. In the general formulation above, where we can consider a stock price, which can either move up to Su or down to Sd (u
> 1, d < 1) in any time period. If the stock price moves up to Su, we suppose that the payoff from the option is Cu; if the stock price moves down to Sd, we suppose the payoff from the derivative is Cd. This situation is illustrated in Figure 3.2.
Figure 3.2 Stock and Option Prices in General Two-step Tree Su2 Cuu
Sud
Cuu
Sd2 Cdd Su
Sd Cu
Cd S
C A
B
C
D
E
F
We create a replicating portfolio consisting of a long position in shares and a short position in one option. We calculate the value of that makes the portfolio risk-free. If there is an up movement in the stock price, the value of the portfolio at the end of the life of the option is (Su - Cu), and the value of the portfolio is (Sd - Cd) if a down movement. The two are equal only when Su - Cu = Sd - Cd⇒
d u
d u
S S
C C
−
= −
∆ . (3.1)
Because of the time value, a risk-free interest rate will be considered in terms that the portfolio is risk-free.
We assume that the time at node B or C in Figure 3.2 is T and the risk-free interest rate is r, then the present value of the portfolio will be (Su - Cu) e-rT.
The cost of setting up the portfolio is: (S - C)
It follows that (Su - Cu)e-rT =S - C ⇒ C = e -rT[pCu + (1- p)Cd] by substituting from equation (3.1) and simplifying, in which
d u
d p e
rT
−
= − .
This is for one-step binomial process. By following the same logic, we can value an option in a two-step (two-period) binomial tree. In the process of a two-period tree, we start with the last time period and move backwards in time until the current point in time. Further, we can, in iterative way, proceed the valuation5 for multi-period binomial tree form backwards in time to the start point. Since the calculation for multi-period binomial tree is too complicated and tedious, thus generating a general formula is rather necessary.
5 See Appendix 1 for better understanding.
Table 3.2 The Binomial Option Pricing Model (Cox, Ross, Rubinstein Option Pricing Model) The General Binomial Option
Pricing Formula
Description of the inputs
C Call option value
è [] Binomial function (distribution)
[anp] E e [anp]
S
C= ∗θ ;, '− ∗ −nθ ;,
Where p
r
p'=u p Risk neutral probability
a Smallest non-negative integer greater than ln (E/Sdn)/ln (u/d)
e Exponential function r Short term interest rate until expiration E Exercise price of option n Number of discrete periods until
expiration
u Possible upward movement d Possible downward movement in prices
Nonetheless, a binomial tree has several curious, and possibly limiting, properties. For example, all sample paths that lead to the same node in the tree have the same risk-neutral probability. The types of volatility – objective, subjective and realized – are indistinguishable; and, in the limit, its continuous-time sample path is not differentiable at any point.
3.3.2 Extending the Binomial Model to Continuous Time—Black-Scholes Option Pricing Model
The binomial pricing model can be extended to derive a continuous time equivalent—Black-Scholes model if we hold the amount of calendar time (one year for example) constant and divide it into more and more binomial tree’s nodes. We will define T as the life of the option expressed as a fraction of a year and will divide T into n smaller time intervals. As n becomes larger, the
